Question

∆Challenge to all BRAINLY USERS
$\frac{ {a}^{ - 1} }{ {a}^{ - 1} + {b }^{ - 1} } + \frac{ {a}^{ - 1} }{ {a }^{ - 1} - {b}^{ - 1} } = \frac{2 {b}^{2} }{ {b}^{2} - {a}^{2} }$
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Answer 1

One can multiply 1 to both fractions on the LHS. The answer is quite simple.

We know:-

$\rightarrow\dfrac{ab}{ab}=1\ (ab\neq0)$

So we are going to multiply 1.

Calculation:-

$\rightarrow1\times\dfrac{a^{-1}}{a^{-1}+b^{-1}}+1\times\dfrac{a^{-1}}{a^{-1}-b^{-1}}$

$=\dfrac{ab}{ab}\times\dfrac{a^{-1}}{a^{-1}+b^{-1}}+\dfrac{ab}{ab}\times\dfrac{a^{-1}}{a^{-1}-b^{-1}}$

$=\dfrac{b}{b+a}+\dfrac{b}{b-a}$

$=\dfrac{b(b-a)+b(b+a)}{(b+a)(b-a)}$

$=\dfrac{b^2-ab+b^2+ab}{b^2-a^2}$

$=\dfrac{2b^2}{b^2-a^2}$

Hence proven.

Answer 2

Step-by-step explanation:

given :

• ∆Challenge to all BRAINLY USERS
• $\frac{ {a}^{ - 1} }{ {a}^{ - 1} + {b }^{ - 1} } + \frac{ {a}^{ - 1} }{ {a }^{ - 1} - {b}^{ - 1} } = \frac{2 {b}^{2} }{ {b}^{2} - {a}^{2} }$

to find :

• $\frac{ {a}^{ - 1} }{ {a}^{ - 1} + {b }^{ - 1} } + \frac{ {a}^{ - 1} }{ {a }^{ - 1} - {b}^{ - 1} } = \frac{2 {b}^{2} }{ {b}^{2} - {a}^{2} }$

solution:

• please check the attached file